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A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).

%I A306705 #25 Sep 08 2022 08:46:21
%S A306705 1,4,6,48,10,576,14,1536,162,1600,22,497664,26,3136,3600,122880,34,
%T A306705 1679616,38,2304000,7056,7744,46,3057647616,750,10816,17496,6322176,
%U A306705 58,3317760000,62,23592960,17424,18496,19600,470184984576,74,23104,24336,23592960000,82
%N A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).
%H A306705 Robert Israel, <a href="/A306705/b306705.txt">Table of n, a(n) for n = 1..10000</a>
%F A306705 a(p) = 2p for p = primes (A000040).
%F A306705 a(n) = (Product_{d|n} tau(d)) * (Product_{d|n} d) = A211776(n) * A007955(n).
%F A306705 From _Robert Israel_, Mar 24 2019: (Start)
%F A306705 a(p^k) = (k+1)! * p^(k*(k+1)/2) for primes p.
%F A306705 a(p*q) = 16*p^2*q^2 if p and q are distinct primes. (End)
%e A306705 a(6) = 1*tau(1) * 2*tau(2) * 3*tau(3) * 6*tau(6) = (1*1) * (2*2) * (3*2) * (6*4) = 576.
%p A306705 f:= proc(n) uses numtheory; local d;
%p A306705   mul(d*tau(d),d = divisors(n))
%p A306705 end proc:
%p A306705 map(f, [$1..100]); # _Robert Israel_, Mar 24 2019
%t A306705 Table[n^(DivisorSigma[0, n]/2) * Product[DivisorSigma[0, k], {k, Divisors[n]}], {n, 1, 60}] (* _Vaclav Kotesovec_, Mar 10 2019 *)
%o A306705 (Magma) [&*[d * NumberOfDivisors(d): d in Divisors(n)]: n in [1..100]]
%o A306705 (PARI) a(n) = my(res = 1); fordiv(n, d, res *= d*numdiv(d)); res; \\ _Michel Marcus_, Mar 06 2019
%Y A306705 Cf. A000005, A060640 (Sum_{d|n} d*tau(d)), A007955, A211776.
%K A306705 nonn
%O A306705 1,2
%A A306705 _Jaroslav Krizek_, Mar 05 2019